Anisotropic Observations in Universes with Nonlinear Inhomogeneity

Abstract

We calculate idealized observational relations, within the general theory of relativity, for an off-center observer in an inhomogeneous spherically symmetric region of the universe. The model allows for arbitrarily large nonlinear deviations from homogeneity, but it incorporates the typical case of small perturbations in the gravitational field, for which linear perturbation theory is usually used. The off-center position of the observer and the size of the inhomogeneous region are unrestricted. We give exact expressions for the Hubble and deceleration parameters measured by the off-center observer. Neglecting local velocities relative to the mean geodesic motion, it follows that there is no dipole in the Hubble parameter, only a quadrupole anisotropy. Number counts are given up to second-order in a redshift power series expansion. The off-center geometry and the matching of the inhomogeneous region to the global homogeneous universe are treated analytically. A simple procedure for calculating the cosmic microwave background radiaton anisotropy is described, which can reproduce the observed dipole with a reasonable choice of parameters.